Rejecting pressure fluctuations induced by string movement in drilling*

2nd IFAC Workshop on Control of Systems Governed 2nd IFAC Control 2ndPartial IFAC Workshop Workshop on Control of of Systems Systems Governed Governed by Differentialon Equations 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations Available online at www.sciencedirect.com by Partial Differential Equations June 13-15, 2016. Bertinoro, Italy by Partial Differential Equations June June 13-15, 13-15, 2016. 2016. Bertinoro, Bertinoro, Italy Italy June 13-15, 2016. Bertinoro, Italy

ScienceDirect

IFAC-PapersOnLine 49-8 (2016) 124–129

Rejecting pressure fluctuations induced Rejecting pressure fluctuations induced Rejecting pressure fluctuations induced  string movement in drilling drilling string movement in in drilling 

by by by

Timm Strecker ∗∗ Ole Morten Aamo ∗∗ Timm Timm Strecker Strecker ∗∗ Ole Ole Morten Morten Aamo Aamo ∗∗ Timm Strecker Ole Morten Aamo ∗ ∗ Department of Engineering Cybernetics, NTNU, Norwegian ∗ of Engineering Cybernetics, NTNU, Norwegian ∗ Department Department of Cybernetics, NTNU, Norwegian University of Science and Technology, Trondheim N-7491, Norway Department of Engineering Engineering Cybernetics, NTNU, Norwegian University of Science and Technology, Trondheim N-7491, Norway University of Science and Technology, Trondheim N-7491, (e-mail: [email protected]; [email protected]). University of Science and Technology, Trondheim N-7491, Norway Norway (e-mail: [email protected]; [email protected]). (e-mail: (e-mail: [email protected]; [email protected]; [email protected]). [email protected]). Abstract: Keeping the pressure within predefined bounds is essential for the safety of managed Abstract: Keeping the predefined bounds is for of Abstract: Keeping the pressure pressure within predefined boundscan is essential essential for the the safety safety of managed managed pressure drilling operations, but within drill string movements induce pressure fluctuations that Abstract: Keeping the pressure within predefined bounds is essential for the safety of managed pressure drilling operations, but drill string movements can induce pressure fluctuations that pressure drilling operations, but drill string movements can induce pressure fluctuations that violate margins. We extend results on disturbance rejection for afluctuations 2 × 2 hyperbolic pressurethese drilling operations, but previous drill string movements can induce pressure that violate these margins. We extend previous results on disturbance rejection for a 2 × 2 hyperbolic violate these margins. We extend previous results on disturbance rejection for a 2 × 2 hyperbolic model themargins. fluid dynamics in previous the borehole. a simulation study, which we violate of these We extend resultsWe on perform disturbance rejection for a 2 ×in 2 hyperbolic model of the fluid dynamics in the borehole. perform aa simulation in which we model dynamics in We perform study, in we illustrate controller performance for heaveWe induced oscillations and study, compare it to the model of of the the fluid fluid dynamics in the the borehole. borehole. We perform a simulation simulation study, in which which we illustrate the controller performance for heave induced oscillations and compare it to the illustrate the the controller performance for heave heave induced induced oscillations and limitations compare it it on to the previously available controller. The stochasticity of sea oscillations waves induces illustrate controller performance for and compare to previously available controller. stochasticity of seatowaves induces the previously available The stochasticity of induces limitations on the achievable Further, The the controller is applied intended stringlimitations movements,on where previously performance. available controller. controller. The stochasticity of sea sea waves waves induces limitations on the achievable performance. Further, the controller is applied to intended string movements, where achievable performance. Further, the controller is applied to intended string movements, where control canperformance. be used to remove conservative limitations on to theintended speed ofstring the movement. achievable Further, the controller is applied movements, where control can be used to remove conservative limitations on the speed of the movement. control control can can be be used used to to remove remove conservative conservative limitations limitations on on the the speed speed of of the the movement. movement. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Distributed-parameter systems, Disturbance rejection, Boundary control, Managed Keywords: Distributed-parameter systems, Keywords: Distributed-parameter systems, Disturbance Disturbance rejection, rejection, Boundary Boundary control, control, Managed Managed pressure drilling, Well control Keywords: Distributed-parameter systems, Disturbance rejection, Boundary control, Managed pressure drilling, Well control pressure drilling, Well control pressure drilling, Well control 1. INTRODUCTION and Aamo (2015) to control the pressure at an arbitrary 1. and Aamo to control the pressure an arbitrary 1. INTRODUCTION INTRODUCTION and Aamo (2015) to the pressure at an position in (2015) the well. However, modelat that 1. INTRODUCTION and Aamo (2015) to control control thetheir pressure atneglects an arbitrary arbitrary in the well. However, their model neglects position in the well. However, their model neglects that Maintaining the pressure in a borehole above the pore position the disturbance affects the system throughout the that doposition in the well. However, their model neglects that Maintaining pressure in borehole above the affects system throughout the doMaintaining the pressure in a borehole above the pore pore the the disturbance disturbance affects the system throughout the they dopressure and the below the fracture pressure of the formation main because fluid sticksthe to the drill string. Further, Maintaining the pressure in aa borehole above the pore the disturbance affects the system throughout the dopressure and below the fracture pressure of the formation main because fluid sticks to the drill string. Further, they pressure and below the fracture pressure of the formation main because because fluid sticks sticksharmonic to the the drill drill string. Further, Further, they is essential forbelow the safety of drilling operations. consider only simplistic oscillations, and neglect pressure and the fracture pressure of the Therefore, formation main fluid to string. they is essential for the of drilling operations. Therefore, harmonic oscillations, and neglect is essential for drilling the safety safety of is drilling operations. Therefore, consider only simplistic harmonic oscillations, and a fluid called mud pumped down through the consider that the only heavesimplistic can actually be measured on the using is essential for the safety of drilling operations. Therefore, consider only simplistic harmonic oscillations, andrigneglect neglect aadrill fluid called drilling mud is pumped down through the that the heave can actually be measured on the rig using fluid called drilling mud is pumped down through the that the heave can actually be measured on the rig string, through the bit and up the annulus. The accelerometers. Thus, the disturbance does not need to a fluid called drilling mud is pumped down through the that the heave can actually be measured on the rig using using drill string, through the bit and up the annulus. The accelerometers. Thus, the disturbance does not need to drill string, through the bittheand and up the theremains annulus.within The accelerometers. accelerometers. Thus, the disturbance disturbance does not not need need to mud string, is designed suchthe thatbit pressure be estimated from pressure and flow measurements, which drill through up annulus. The Thus, the does to mud is designed such that the pressure remains within be estimated from pressure and flow measurements, which mud is designed such that the pressure remains within be estimated from pressure and flow measurements, which tolerable margins. such For wells the pressure margins are be improves the from performance theflow observer significantly. mud is designed thatwhere the pressure remains within estimated pressureofand measurements, which tolerable margins. For where pressure are the performance of the the observer significantly. tolerable margins. For wells wells where the the pressure margins are improves improves the of significantly. tight, so-called managed pressure drilling wasmargins developed, String movements when tripping drill string, or placing tolerable margins. For wells where the pressure margins are improves the performance performance of the the observer observer significantly. tight, so-called managed pressure drilling was developed, String movements when tripping the drill string, or placing tight, so-called managed pressure drilling was developed, String movements when tripping the drill string, enabling fast active control of the annular pressure using a casing in the borehole, are actually more common, as they tight, so-called managed pressure drilling was developed, String movements when tripping the drill string, or or placing placing enabling fast active control of the annular pressure using a casing in the borehole, are actually more common, as enabling fast active control control of the the annular pressure using aa casing casing also in the the borehole, are actually more common, as they they choke andfast backpressure pump at the outletpressure of the annulus, occur when drillingare from fixed more platforms. A common enabling active of annular using in borehole, actually common, as they choke and backpressure pump at the outlet of the annulus, occur also when drilling from fixed platforms. A common choke and backpressure pump at the outlet of the annulus, occur also when drilling from fixed platforms. A common without to change theatmud properties procedure is to calculate the fixed maximum velocity that rechoke andhaving backpressure pump the outlet of the(Godhavn annulus, occur also when drilling from platforms. A common without having to the (Godhavn is to calculate maximum velocity that rewithout having However, to change changedrill the mud mud properties (Godhavn procedure is the maximum velocity that et al. (2010)). stringproperties movements disturb procedure sults in tolerable pressurethe fluctuation without having to change the mud properties (Godhavn procedure is to to calculate calculate the maximum(Mitchell velocity (1988)). that rereet al. (2010)). However, drill string movements disturb sults in tolerable pressure fluctuation (Mitchell (1988)). et al.system (2010)). However, drill string movements movements disturb sults tolerable pressure (Mitchell theal. andHowever, must bedrill compensated for. Drilldisturb string Active control couldfluctuation increase the allowable(1988)). speed, et (2010)). string sults in inpressure tolerable pressure fluctuation (Mitchell (1988)). the system and Drill string pressure the system occur and must must be compensated for. Drillmotion string Active Active time. pressure control control could could increase increase the the allowable allowable speed, speed, movements due tobe thecompensated wave-inducedfor. heaving saving the system and must be compensated for. Drill string Active pressure control could increase the allowable speed, movements occur due to the wave-induced heaving motion saving time. movements occur due to the wave-induced heaving motion saving time. of a floatingoccur drilling andwave-induced during tripping. Tripping is saving The paper movements duerig, to the heaving motion time.is organized as follows. In Section 2, we introof floating rig, during tripping. is is organized as follows. Section 2, we introof floating drilling drilling rig, and and during tripping. Tripping is The The paper is as In Section 2, we theaaa intentional movement of the string into orTripping out of the duce paper the model, and in Section 3 weIn of floating drilling rig, and during tripping. Tripping is The paper is organized organized as follows. follows. Inpresent Sectionthe 2, controller we introintrothe intentional movement of the string into or out of the duce the model, and in Section 3 we present the controller the intentional movement of the the string into or out out of of the the duce model, and Section present well,intentional e.g. in order to change the string drilling equipment. We design. relevant are controller discussed the movement of into or duce the theThree model, and in in simulation Section 33 we wescenarios present the the controller well, e.g. in order the drilling We Three well, e.g.the in attenuation order to to change change thestring-movement drilling equipment. equipment. We design. design. Three relevant simulation scenarios are discussed consider of such induced in Section 4. relevant Finally, simulation concluding scenarios remarks are are discussed given in well, e.g. in order to change the drilling equipment. We design. Three relevant simulation scenarios are discussed consider the attenuation of such string-movement induced in Section 4. Finally, concluding remarks given in consider the attenuation of such string-movement induced in Section 4. Finally, concluding remarks are pressure the fluctuations using the choke and measurements Section 5. 4. Finally, concluding remarks are consider attenuation of such string-movement induced in Section are given given in in pressure fluctuations using the choke and measurements Section 5. pressure fluctuations using the choke and measurements Section 5. that are available on the rig. This is one of several appressure fluctuations using the choke and measurements Section 5. that are on rig. is apthat are available available on the thewhere rig. This This is one one of of several several applications within drilling the distributed dynamics that are available on the rig. This is one of several ap2. MODELING plications within drilling where the distributed dynamics plications within drilling where the distributed dynamics 2. of the system must be taken intothe account (Di Meglio and plications within drilling where distributed dynamics 2. MODELING MODELING 2. MODELING of the must of the system system must be be taken taken into into account account (Di (Di Meglio Meglio and and Aarsnes (2015)). of the system must be taken into account (Di Meglio and We consider the following model for the annular pressure Aarsnes (2015)). Aarsnes (2015)). While drilling from a floating rig, a mechanical system We Aarsnes (2015)). We consider the following following model model for for the annular annular pressure consider the and flow: While drilling from a floating rig, a mechanical system We consider the following model for the the annular pressure pressure While drilling from movement floating from rig, aathe mechanical system and flow: decouples the rig’s string. However, While drilling from aa floating rig, mechanical system and flow: β decouples the rig’s movement from the string. However, and flow: decouples the rig’s movement from the string. However, every 30 mthe therig’s drillmovement string must fixed to the rig in decouples frombethe string. However, pt (z, t) = − β q (z, t) (1) β every 30 the must be to in β qqzz (z, p t) (1) every 30 m m the drill drill string mustsegment, be fixed fixedand to the the rig rig in order to extend it bystring another string pttt (z, (z, t) t) = =− −A (z, t) (1) every 30 m the drill string must be fixed to the rig in z p (z, t) = − q (z, t) (1) A z order to extend it by another segment, and the string A F F d order to extend it by another segment, and the string A moves with the rig. The distributed dynamics of the heave order to extend it by another segment, and the string A F F v (z, t) = − p (z, t) − q(z, t) + (z, t) − Ag (2) q A F moves with the rig. The dynamics the Fρddd vvdd (z, moves with thefirst rig. addressed The distributed distributed dynamics of the heave heave pzz (z, q(z, t) + +F qqttt (z, ρp ρ q(z, problem were in Landet et al. of (2013), and moves with the rig. The distributed dynamics of the heave (z, t) t) = =− −A (z, t) t) − −F (z, t) t) − − Ag Ag (2) (2) vdd (z, t) = − t) − t) + ρ t) − Ag (2) qt (z, ρ pzz (z, ρ q(z, t) problem were first addressed in Landet et al. (2013), and ρ ρ ρ problem were first addressed in Landet et al. (2013), and a controller based on a discretized model. ρ ρd vd (0, t) ρ q(0, t) = −A (3) problem werewas firstdesigned addressed in Landet et al. (2013), and aaThe controller was designed based on a discretized model. q(0, t) = −A vvdd (0, t) (3) d controller was designed designed based on on discretized model. q(0, t) = −A (0, t) (3) heave disturbance is modeled as aaandiscretized inflow at the bot- where d a controller was based model. q(0, t) = −A v (0, t) z ∈ [0, l],dt d≥ 0, l is the length of the well, p(z, t)(3) is The heave disturbance is modeled as an inflow at the botThe heave disturbance is modeled as an inflow at the botwhere z ∈ [0, l], t ≥ 0, l is the length of the well, p(z, t) is tom. Most relevant to this work, a backstepping controller The heave disturbance is modeled as an inflow at the bot- where zz ∈ [0, l], ttthe ≥ 0, ll is the length of the well, p(z, t) is pressure, q(z, t) volumetric flow rate, v (z, t) is the where ∈ [0, l], ≥ 0, is the length of the well, p(z, t) is d tom. Most relevant to this work, a backstepping controller tom. Most relevant tohyperbolic this work, work, aaPDEs backstepping controller pressure, q(z, t) the volumetric flow rate, v (z, t) is the basedMost on the 2 × 2to was developed in drill tom. relevant this backstepping controller d pressure, q(z, t) the volumetric flow rate, v (z, t) is d string velocity, A the cross sectional area of the pressure, q(z, t) the volumetric flow rate, vd (z, t) is based on the 2 × 2 hyperbolic PDEs was developed in based on the 2 × 2 hyperbolic PDEs was developed in A cross sectional area of Aamo (2013), method was generalized in Anfinsen based on the and 2 ×this 2 hyperbolic PDEs was developed in drill drill string string velocity, A the the cross by sectional area of the the Ad velocity, is the area replaced the drill string, β drill string velocity, A the cross sectional area of the Aamo Aamo (2013), (2013), and and this this method method was was generalized generalized in in Anfinsen Anfinsen annulus, annulus, A is the area replaced by the drill string, β Aamo (2013), and this method was generalized in Anfinsen d  annulus, A is the area replaced by the drill string, β d is ρthe bulk modulus, density, g the gravitational constant, F Financial support from Statoil ASA is gratefully acknowledged. annulus, A area replaced by the drill string, β d  bulk modulus, ρ density, gg the gravitational constant, F support from Statoil ASA is gratefully acknowledged.  bulk modulus, ρ density, the gravitational constant, F Financial support from Statoil ASA is gratefully acknowledged.  Financial bulk modulus, ρ density, g the gravitational constant, F Financial support from Statoil ASA is gratefully acknowledged. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 International Federation of 125Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 International Federation of 125 Peer review© under responsibility of International Federation of Automatic Copyright © 2016 International Federation of 125 Automatic Control Copyright © 2016 International Federation of 125Control. Automatic Control 10.1016/j.ifacol.2016.07.429 Automatic Control Automatic Control

125 125 125 125

IFAC CPDE 2016 June 13-15, 2016. Bertinoro, Italy

Timm Strecker et al. / IFAC-PapersOnLine 49-8 (2016) 124–129

and Fd are friction factors, and subscripts z and t denote derivatives with respect to space and time, respectively. The parameters A, β, ρ, F , and Fd can vary with z, but we omit this dependency for the sake of readability. The control objective is to control the bottomhole pressure to a given setpoint psp , i.e. (4) p(0, t) = psp . The topside boundary condition is left as a control input. Model (1)-(2) for the annular dynamics was used in Aarsnes et al. (2013) for studying resonances and is an extension of the model that was used in Aamo (2013). The last term on the right-hand side of momentum balance (2) is added to model the “mud-clinging” effect: like in Couette flow, fluid sticks to the drill string, and via the viscous forces in the fluid the moving string induces a momentum on the fluid. We assume linear friction, which was shown to be reasonable for laminar flow of Newtonian fluids, and a limited frequency range in case of oscillations (Stecki and Davis (1986a), Stecki and Davis (1986b)). Thus, the drill string movement/disturbance enters the system not only as the inflow at the bottom of the hole, i.e. through a boundary condition, but also throughout the domain. The significance of the mud-clinging effect is illustrated in Section 4.1. Simulations of the model in Aarsnes et al. (2013), which also contains drill string elasticity, have shown that the drill string is approximately rigid in most situations, unless the string’s resonance frequency is excited for an extended period. Therefore, we assume that the velocity of the drill string equals the velocity at the rig, vd (z, t) = vd (l, t) ∀z ∈ [0, l], in which case the drill string velocity can be obtained from measurements on the rig. Thus, the estimation problem is significantly simplified compared to Aamo (2013). 2.1 Disturbance model Due to the length of the well and the hyperbolic nature of the model, there is a delay between topside actuation and downhole effect. Therefore, the control involves feedforward using a prediction of the disturbance. This will be made precise in Section 3.2. We consider two distinct cases of string movement. First, the string movement is intentional. In this case, the velocity is determined by the operator, i.e. it is known a priori. Second, we consider heave-induced oscillations. Here, the string velocity depends on the waves and the drilling rig’s response. For this case, we consider the linear model ˙ ¯ X(t) = AX(t), (5) ¯ (6) vrig (t) = CX(t), with     0 ωn 0 ω1 ¯ ,... , (7) A = diag −ω1 0 −ωn 0 C¯ = [0 1 . . . 0 1] (8) and vd (l, t) = vrig (t). X(t) can be estimated from measurements of vrig using the observer ˆ˙ ˆ + L (vrig (t) − vˆrig (t)) , (9) X(t) = A¯X(t) ¯ ˆ vˆrig (t) = C X(t), (10) where the observer gain L is chosen such that A¯ − LC¯ is Hurwitz. Heave prediction by this model is imperfect 126

125

because sea waves are stochastic, and the n chosen frequencies ω1 , . . . , ωn do not cover the wave spectrum completely. However, we will assume that it suffices for obtaining short term predictions. 2.2 State transformation In order to use previous results on backstepping controller design, it is desirable to bring system (1)-(3) on diagonal form. In Aamo (2013), Lemma 10, the coordinate transformation   A 1 q(xl, t) + √ (p(xl, t) − psp + ρglx) u(x, t) = 2 βρ (11) lF √

x

× e 2 βρ ,   A 1 q(xl, t) − √ (p(xl, t) − psp + ρglx) v(x, t) = 2 βρ (12) lF − √

x

× e 2 βρ , where x ∈ [0, 1], was given. In the presence of the additional term in (2) we get ut (x, t) = −1 (x)ux (x, t) + c1 (x)v(x, t) + d1 (x, t) (13) vt (x, t) = 2 (x)vx (x, t) + c2 (x)u(x, t) + d2 (x, t) (14) u(0, t) = qv(0, t) + d(t) (15) v(1, t) = U (t) (16) with  1 β , d(t) = −Ad vd (0, t), (17) 1 = 2 = l ρ F √lF x F − √lF x c2 (x) = − e βρ , (18) c1 (x) = − e βρ , 2ρ 2ρ Fd γx Fd −γx d1 (x, t) = e vd (l, t), d2 (x, t) = e vd (l, t), (19) 2ρ 2ρ lF and q = −1. U is the control input to be γ = √ 2

βρ

designed. The control objective becomes u(0, t) = rv(0, t) (20) with r = 1. For (1)-(2), the control law is realized by lF √ A (21) ql (t) = √ (pl (t) − psp + ρgl) + 2U (t)e 2 βρ . βρ The first term is in the form of a “passive” boundary condition and is responsible for avoiding reflections of pressure waves, while the latter term involving U is the active control input. The model for vd remains unchanged. In case of a linear model of the form (5)-(6), the disturbance model is ˙ X(t) = AX(t), (22) d(t) = CX(t), d1 (x, t) = C1 (x)X(t), d2 (x, t) = C2 (x)X(t), ¯ C = −Ad C, ¯ C1 (x) = where A = A, lF − √ x 2 βρ

(23) (24) (25) Fd 2 2ρ e

lF √

βρ

x

¯ and C,

C¯ in our case, and X can be estimated C2 (x) = F2ρd e using (9) - (10). More generally, the method in this paper can also be applied for any nonlinear disturbance model that provides a prediction of the disturbance. The requirements on the prediction are made precise in Section 3.2.

IFAC CPDE 2016 126 June 13-15, 2016. Bertinoro, Italy

Timm Strecker et al. / IFAC-PapersOnLine 49-8 (2016) 124–129

3. CONTROLLER DESIGN 3.1 Backstepping transformation The backstepping transformation  x K uu (x, ξ)u(ξ, t)dξ α(x, t) =u(x, t) − 0  x uv − K (x, ξ)v(ξ, t)dξ,

(26)

0

 x K vv (x, ξ)v(ξ, t)dξ β(x, t) =v(x, t) − 0  x − K vu (x, ξ)u(ξ, t)dξ

(27)

was introduced in Vazquez et al. (2011) to map the system without disturbances into a target system, where the kernels satisfy the partial differential equations 1 (x)Kxuu (x, ξ) + 1 (ξ)Kξuu (x, ξ) (28) = −1 (ξ)K uu (x, ξ) − c2 (ξ)K uv (x, ξ) 1 (x)Kxuv (x, ξ) − 2 (ξ)Kξuv (x, ξ) (29) = 2 (ξ)K uv (x, ξ) − c1 (ξ)K uu (x, ξ) 2 (x)Kxvu (x, ξ) − 1 (ξ)Kξvu (x, ξ) (30) = 1 (ξ)K vu (x, ξ) + c2 (ξ)K vv (x, ξ) 2 (x)Kxvv (x, ξ) + 2 (ξ)Kξvv (x, ξ) (31) = −2 (ξ)K vv (x, ξ) + c1 (ξ)K vu (x, ξ) on T = {(x, ξ) : 0 ≤ ξ ≤ x ≤ 1} and boundary conditions 2 (0) uv K uu (x, 0) = K (x, 0) (32) q1 (0) c1 (x) K uv (x, x) = (33) 1 (x) + 2 (x) c2 (x) K vu (x, x) = − (34) 1 (x) + 2 (x) q1 (0) vu K vv (x, 0) = K (x, 0). (35) 2 (0) In the presence of the disturbances, and using the control input  1 U (t) = V (t) + K vu (1, ξ)u(ξ, t) + K vv (1, ξ)v(ξ, t)dξ, 0

(36) transformation (26)-(27) maps the system (13)-(16) into αt (x, t) = − 1 (x)αx (x, t) − 1 (0)K uu (x, 0)d(t)  x + d1 (x, t) − K uu (x, ξ)d1 (ξ, t)dξ (37) 0  x − K uv (x, ξ)d2 (ξ, t)dξ, 0

(38)

0

(44) the solution to (38) satisfies t fβ (1 − h−1 β(0, t) − β(1, t − dα ) = β (t − γ), γ)dγ. (45) t−dα

For the linear disturbance model (22)-(25), fβ becomes linear in the disturbance state X(t), and there is a separation between space and time-varying terms:  fβ (x, t) = − 1 (0)K vu (x, 0)C + C2 (x)   x vu vv K (x, ξ)C1 (ξ) + K (x, ξ)C2 (ξ)dξ X(t) − 0

=:Kβ (x)X(t).

(46) Using this, the analogue to (45) is  dβ −Aτ β(0, t) − β(1, t − dβ ) = Kβ (1 − h−1 dτ X(t). β (τ ), 0)e 0

(47) Similar calculations can be done for the α-system, but they are not required here. 3.2 Controller design Using (39), (40), and (45), we get α(0, t) =qβ(0, t) + d(t) =rβ(0, t) + (q − r)β(0, t) + d(t) =rβ(0, t) + (q − r)V (t − dβ ) + d(t)  t + (q − r) fβ (1 − h−1 β (t − γ), γ)dγ.

(48)

t−dβ

The control objective (41) is achieved if and only if (setting the sum of all but the first term on the right-hand side zero and shifting time) 1 d(t + dβ ) V (t) = − q−r  t+dβ (49) −1 + fβ (1 − hβ (t + dβ − γ), γ)dγ. t

0

α(0, t) = qβ(0, t) + d(t), β(1, t) = V (t),

fβ (x, t) = −1 (0)K vu (x, 0)d(t) + d2 (x, t)  x  x − K vu (x, ξ)d1 (ξ, t)dξ − K vv (x, ξ)d2 (ξ, t)dξ, 0

0

βt (x, t) =2 (x)βx (x, t) − 1 (0)K vu (x, 0)d(t)  x + d2 (x, t) − K vu (x, ξ)d1 (ξ, t)dξ 0  x − K vv (x, ξ)d2 (ξ, t)dξ,

which can be shown by following the steps in the proof of Lemma 1 in Aamo (2013). Since α(0, t) = u(0, t) and β(0, t) = v(0, t), the control objective becomes α(0, t) = rβ(0, t). (41) Using Lemma 2 in Aamo (2013), the explicit solution to (37) and (38) can be obtained. With  1 1 hα (z) = dγ, dα = hα (0), (42)  (γ) 1 z  1 1 dγ, dβ = hβ (0), (43) hβ (z) =  (1 − γ) 2 z

(39) (40) 127

Thus, the control input at time t requires knowledge of the disturbances at all times τ ∈ [t, t + dβ ], i.e. dβ into the future. This can be implemented using a disturbance prediction as discussed in Section 2. For the linear disturbance

IFAC CPDE 2016 June 13-15, 2016. Bertinoro, Italy

Timm Strecker et al. / IFAC-PapersOnLine 49-8 (2016) 124–129

model, V can be implemented as (using the predicted ˜ + τ ) = eAτ X(t)) ˆ disturbance X(t 1 ˆ CeAdβ X(t) V (t) = − q−r  dβ (50) A(dβ −τ ) ˆ − Kβ (1 − h−1 (τ ))e dτ X(t). β 0

As mentioned earlier, there can be a mismatch between predicted and actual disturbances, resulting in an error in the control objective. For the linear case, inserting (50) into (48) yields   ˆ − dβ ) u(0, t) =rv(0, t) + d(t) − CeAdβ X(t  t + (q − r) fβ (1 − h−1 β (t − γ), γ)dγ t−dβ

−



dβ 0

Kβ (1 −

A(dβ −τ ) ˆ h−1 dτ X(t β (τ ))e



− dβ ) .

(51) This illustrates how a prediction error in the disturbance imposes limitations on the achievable controller performance.

127

The friction factor F was calculated using a modification of the Hagen-Poisseuille law for the annular geometry and viscosity µ , and Fd = 0.5AF . The length l varies in the examples. The pressure setpoint was set to a value slightly above the hydrostatic pressure, psp = 1.05ρgl, but this does not affect the dynamics. 4.1 Mud-clinging effect In this example, we compare the controller (50) with the controller from Aamo (2013) in the presence of the mud-clinging effect. In order to isolate the effect of the additional momentum term, we assume a sinusoidal (i.e. predictable) heave motion with period 12 s and amplitude 1 m, vrig = 2πa sin(ωt) (ω = 2π 12 and a = 1), and use l = 2000 m. In Figure 1, the downhole pressure regulation error p(0, t) is depicted for three cases: controller (50), the controller from Aamo (2013), and U (t) ≡ 0 for comparison. The controller neglecting the mud-clinging effect achieves a clear reduction compared to no control, but significant pressure fluctuations remain.

3.3 Observer To implement (36), u and v are required throughout the domain. In practice, only measurements at the right boundary are available. Therefore, we use the observer ux (x, t) + c1 (x)ˆ v (x, t) + d1 (x, t) u ˆt (x, t) = − 1 (x)ˆ + p1 (x)(u(1, t) − u ˆ(1, t)) (52) vx (x, t) + c2 (x)ˆ u(x, t) + d2 (x, t) vˆt (x, t) = 2 (x)ˆ + p2 (x)(u(1, t) − u ˆ(1, t)) (53) u ˆ(0, t) = qˆ v (0, t) + d(t) (54) vˆ(1, t) = U (t), (55) where p1 and p2 are output injection gains to be designed. Forming error equations by subtracting (52)-(55) from (13)-(16) yields the error system ux (x, t) + c1 (x)˜ v (x, t) − p1 (x)˜ u(1, t) u ˜t (x, t) = − 1 (x)˜ (56) v˜t (x, t) = 2 (x)˜ vx (x, t) + c2 (x)˜ u(x, t) − p2 (x)˜ u(1, t) (57) u ˜(0, t) = q˜ v (0, t) (58) v˜(1, t) = 0. (59) Note that the disturbance terms cancel. Therefore, the observer in Vazquez et al. (2011) can be applied. The observer error becomes zero within dα + dβ . The output injection gains p1 and p2 are obtained by solving observer kernel PDEs similar to (28)-(35). They are given in Vazquez et al. (2011). 4. SIMULATIONS In this section, we present examples to illustrate the performance of the controller. The following common parameters are used β = 1.6 × 109 Pa, Ad = 0.0127 m2 , A = 0.0198 m2 ,

3

ρ = 1420 kg/m , µ = 42 mPas, F = 700 kg/m3 s, 128

Fig. 1. Downhole pressure oscillations using controller (50), the controller from Aamo (2013), and using no control (U = 0). 4.2 Real heave data Heave measurements from a semisubmersible rig are depicted in Figure 2. This is a typical heave spectrum for a rig operated in the North Sea, where high-frequency components are dampened by the rig. The heave velocity is estimated using (9) - (10), where the frequencies ω1 , . . . , ω5 are chosen according to the velocity’s frequency spectrum, see Figure 3. In practice, the spectrum can be updated regularly to match the current sea state. A time series of the resulting pressure oscillations in a 5000 m long well is depicted in Figure 4. The controller succeeds in reducing the pressure fluctuations at the bottom of the well, but some oscillations remain due to the error in the disturbance prediction. Motivated by (51), measured and predicted rig velocities are compared in Figure (2), where the predicted velocity is given by ¯ ˆ ¯ AT v˜rig (t) = Ce X(t − T ) (60)

IFAC CPDE 2016 June 13-15, 2016. Bertinoro, Italy 128

Timm Strecker et al. / IFAC-PapersOnLine 49-8 (2016) 124–129

Fig. 2. Measured heave velocity and predicted velocity using (60) for T = 3, 6 s. for the prediction horizon T . The cummulative distribution functions of the downhole pressure oscillations for different well lengths are depicted in Figure 5. Clearly, the pressure variations increase with well length, because of the longer prediction. Note that these values can be computed directly from Equation (51). The distribution function without active control (U = 0), which is almost independent from l, is depicted for comparison. Considering that uncertainties and practical challenges in the implementation of controller (50) increase the pressure variations, one needs to assess the advantages compared to the simpler strategy U = 0 for long wells. The root mean square as another measure of the downhole pressure variation is depicted in Figure 6. More relevant in practice is the maximum deviation from psp . The figure shows that the maximum varies little with l and attains already a high value at short lengths. However, maxima over a short time might have little effect due to the plasticity of the formation. Therefore,  1 T (|p(0, t) − psp | − 2.5) × 1|p(0,t)−psp |>2.5 dt (61) T 0 can be used as a measure for the “energy” per time beyond the margin psp ± 2.5 bar that is applied to the formation. The figure shows that this energy increases significantly for l > 6000 although the maximum remains approximately constant. Figures 5 and 6 can be used to judge if the pressure oscillations are tolerable based on the expected, or currently observed, heave motion.

Fig. 3. Heave velocity spectrum and chosen frequencies.

Fig. 4. Pressure deviation from steady conditions for real heave data and l = 5000 m.

4.3 Intended drill string movements Another interesting scenario is when running the drill string, or casing, into or pulling it out of the borehole. A common procedure is to estimate the maximum velocity that results in a tolerable surge or swab pressure, respectively, see e.g. Mitchell (1988). Control could be used to actively attenuate these pressure fluctuations, allowing a higher tripping speed. This case serves as an example where the string velocity is known a priori. The controller in this paper is designed on the assumption that string movement causes a flow at the bottom. This assumption is satisfied at least for the lowermost string segments. 129

Fig. 5. Cummulative distribution function of the downhole pressure regulation error for different well lengths and U = 0 for comparison.

IFAC CPDE 2016 June 13-15, 2016. Bertinoro, Italy

Timm Strecker et al. / IFAC-PapersOnLine 49-8 (2016) 124–129

129

The method was applied to pressure control in drilling. The control law requires a prediction of the disturbance that can be obtained either from a schedule of an intentional string movement or from a model. Limits on controller performance due to unpredictability of the disturbance have been investigated. Further work could include improving the prediction of the velocity in the heave problem, e.g. by heave measurements in some distance around the rig. The drill string dynamics could be included in the model and controller design to relax the assumption of rigidity. For this purpose, results in Hu et al. (2015) might be exploited. To allow control when the string is far off bottom, the controller could be modified to allow inflow within the domain. Finally, uncertainty in bulk modulus and friction parameters should be explored. REFERENCES Fig. 6. Root mean square, maximum value and “energy” as defined in (61) of the downhole pressure regulation error over well length.

Fig. 7. Bottomhole pressure when pulling the drill string with and without control. For a given string velocity, pressure fluctuations increase with decreasing clearance between drill string and borehole wall. The clearance is smallest in deep parts of a well, because the well diameter decreases with each casing that is installed. Also, the bottom is the most pressure sensitive part of a well, because the upper parts have been protected by casing and cement. Therefore, the most relevant case is when the bit is relatively close to the bottom, see also Mitchell (2004). Judging how far the string may be off bottom is beyond the scope of this paper. Figure 7 shows the bottomhole pressure where the drill string is pulled by one segment (30m) with a maximum velocity of 1 m/s. The pressure deviation without control would be too severe in certain cases, while the backstepping controller attenuates all pressure fluctuations. 5. CONCLUSIONS We have extended previous results on backstepping control of 2 × 2 linear hyperbolic PDEs to attenuate disturbances appearing at the boundary and throughout the domain. 130

Aamo, O.M. (2013). Disturbance rejection in 2 x 2 linear hyperbolic systems. IEEE Transactions on Automatic Control, 58(5), 1095–1106. Aarsnes, U.J.F., Aamo, O.M., Hauge, E., and Pavlov, A. (2013). Limits of controller performance in the heave disturbance attenuation problem. In 2013 European Control Conference (ECC), 1071–1076. Anfinsen, H. and Aamo, O.M. (2015). Disturbance rejection in the interior domain of linear 2 2 hyperbolic systems. IEEE Transactions on Automatic Control, 60(1), 186–191. Di Meglio, F. and Aarsnes, U. (2015). A distributed parameter systems view of control problems in drilling. In 2nd IFAC Workshop on Automatic Control in Offshore Oil and Gas Production, Florian´ opolis, Brazil. Godhavn, J.M. et al. (2010). Control requirements for automatic managed pressure drilling system. SPE Drilling & Completion, 25(03), 336–345. Hu, L., Di Meglio, F., Vazquez, R., and Krstic, M. (2015). Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes. arXiv preprint arXiv:1504.07491. Landet, I.S., Pavlov, A., and Aamo, O.M. (2013). Modeling and control of heave-induced pressure fluctuations in managed pressure drilling. IEEE Transactions on Control Systems Technology, 21(4), 1340–1351. Mitchell, R. (1988). Dynamic surge/swab pressure predictions. SPE drilling engineering, 3(03), 325–333. Mitchell, R.F. (2004). Surge pressures in low-clearance liners. In 2004 IADC/SPE Drilling Conference. Society of Petroleum Engineers. Stecki, J. and Davis, D. (1986a). Fluid transmission lines-distributed parameter models part 1: A review of the state of the art. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 200(4), 215–228. Stecki, J. and Davis, D. (1986b). Fluid transmission lines-distributed parameter models part 2: comparison of models. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 200(4), 229–236. Vazquez, R., Krstic, M., and Coron, J.M. (2011). Backstepping boundary stabilization and state estimation of a 2× 2 linear hyperbolic system. In 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 4937–4942.